Hey, everyone. Welcome back. So in another video, we looked at how to use the substitution method to solve a system of linear equations. The idea was that we would take a variable and isolate it like x equals 2, and we would plug it into another equation to get rid of one of the variables and make the equation simpler. This just becomes y=5×2−3, which we can solve. This just ends up being y=7, and then we would solve it that way,x=2,y=7. In this video, I'm going to show you a different method called the elimination method of getting rid of one of the variables to make your equations simpler. Alright? So it involves a few different steps. I'm going to break it down for you and show you that it's really not so bad. But the gist of it is that if I wanted to solve a system of equations like this, then I could use the substitution method. I absolutely could. But I'm going to show you a different but much faster way to solve these kinds of problems. The basic gist of it is that instead of substituting, we're actually going to add the equations together. What do I mean by that? So we've added variables, and we've added, like, expressions before. Adding equations is actually pretty straightforward. You line up the equations, and then you would actually just add the coefficients of the x's and y's and the numbers just straight down. So you just add everything top to bottom. What you'll see here is that 1 and negative one actually will cancel out. It'll get rid of that x variable just like we got rid of it in the substitution method.
The y's will combine. We'll get 2y, and then 1 and 5 becomes 6. Now this equation is much easier to solve. It's just y equals 3. And then from here, we can actually solve the rest of the problem by plugging it back into one of the other equations. So the whole idea is that we want to add the equations together because we want to eliminate one of the variables. That's why we call it the elimination method. So I just want to point out here again, you're going to use this method. It's generally good to use this method whenever your equations are already in standard form because you can do stuff like adding the equations together or if they have large coefficients. Alright? Now problems won't always be this simple, so I'm actually going to break it down for you step by step to show you how to do this. Let's go ahead and get started.
So we have another example over here. We've got 3x+2y=1 and −x+y=3. Let's take a look at the first step here. The first step is you're going to want to write both of the equations in standard form in case they aren't already, and you want to align the coefficients vertically on top of each other. Basically, what that means here is you want to line up the equation so that x's are on top of x's, y's on y's, and then constants on constants. So this first step is already done for us because we have these equations in standard form. So the next thing we want to do is just start the process of adding these things together. Now if I try to do this right now, we're going to see is if you try to add these equations together, you're going to get 3x−x=2x, 2y+y=3y, and 1+3=4. So unlike what happened above, we didn't get rid of one of the variables. And that's because we didn't have x's that canceled out or y's that canceled out or anything like that. Alright? So we can't automatically just go ahead and start adding these equations.
What we're going to have to do here is we're going to have to multiply one of the equations or both of them by some number. It could be positive or negative. And the whole goal here is that we do want those x's and y's to cancel out. And the way that they do that...